Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance

Influence of inhibitory autapses on synchronization of inhibitory network gamma oscillations

A recent experimental study showed that inhibitory autapses favor firing synchronization of parvalbumin interneurons in the neocortex during gamma oscillations. In the present paper, to provide a comprehensive and deep understanding to the experimental observation, the influence of inhibitory autapses on synchronization of interneuronal network gamma oscillations is theoretically investigated. Weak, middle, and strong synchronizations of a globally inhibitory coupled network composed of Wang-Buzsáki model without autapses appear at the bottom-left, middle, and top-right of the parameter plane with the conductance (gsyn) and the decay constant (τsyn) of inhibitory synapses taken as the x-axis and y-axis, respectively. After introducing inhibitory autapses, the border between the strong and middle synchronizations in the (gsyn, τsyn) plane moves to the top-right with increasing the conductance (gaut) and the decay constant (τaut) of autapses, due to that interspike interval of the single neuron becomes longer, leading to that larger τsyn is needed to ensure the strong synchronization. Then, the synchronization degree of middle and strong synchronizations around the border in the (gsyn, τsyn) plane decreases, while of strong synchronization in the remaining region remains unchanged. The synchronization degree of weak synchronization increases with increasing τaut and gaut, due to that the inhibitory autaptic current becomes strong and long to facilitate synchronization. The enhancement of weak synchronization modulated by inhibitory autapses is also simulated in the random, small-world, and scale-free networks, which may provide explanations to the experimental observation. These results present complex dynamics of synchronization modulated by inhibitory autapses, which needs future experimental demonstrations.

Increased habitat connectivity induces diversity via noise-induced symmetry breaking

Stochasticity or noise is omnipresent in ecosystems that mediates community dynamics. The beneficial role of stochasticity in enhancing species coexistence and, hence, in promoting biodiversity is well recognized. However, incorporating stochastic birth and death processes in excitable slow-fast ecological systems to study its response to biodiversity is largely unexplored. Considering an ecological network of excitable consumer-resource systems, we study the interplay of network structure and noise on species' collective dynamics. We find that noise drives the system out of the excitable regime, and high habitat patch connectance in the ordered as well as random networks promotes species' diversity by inducing new steady states via noise-induced symmetry breaking.

Synchronization and Enhanced Catalysis of Mechanically Coupled Enzymes

We examine the stochastic dynamics of two enzymes that are mechanically coupled to each other, e.g., through an elastic substrate or a fluid medium. The enzymes undergo conformational changes during their catalytic cycle, which itself is driven by stochastic steps along a biased chemical free energy landscape. We find conditions under which the enzymes can synchronize their catalytic steps, and discover that the coupling can lead to a significant enhancement in their overall catalytic rate. Both effects can be understood as arising from a global bifurcation in the underlying dynamical system at sufficiently strong coupling. Our findings suggest that, despite their molecular scale, enzymes can be cooperative and improve their performance in metabolic clusters.

Coherence resonance in neuronal populations: Mean-field versus network model

The counterintuitive phenomenon of coherence resonance describes a nonmonotonic behavior of the regularity of noise-induced oscillations in the excitable regime, leading to an optimal response in terms of regularity of the excited oscillations for an intermediate noise intensity. We study this phenomenon in populations of FitzHugh-Nagumo (FHN) neurons with different coupling architectures. For networks of FHN systems in an excitable regime, coherence resonance has been previously analyzed numerically. Here we focus on an analytical approach studying the mean-field limits of the globally and locally coupled populations. The mean-field limit refers to an averaged behavior of a complex network as the number of elements goes to infinity. We apply the mean-field approach to the globally coupled FHN network. Further, we derive a mean-field limit approximating the locally coupled FHN network with low noise intensities. We study the effects of the coupling strength and noise intensity on coherence resonance for both the network and the mean-field models. We compare the results of the mean-field and network frameworks and find good agreement in the globally coupled case, where the correspondence between the two approaches is sufficiently good to capture the emergence of coherence resonance, as well as of anticoherence resonance.

Canard resonance: on noise-induced ordering of trajectories in heterogeneous networks of slow-fast systems

We analyse the dynamics of a network of semiconductor lasers coupled via their mean intensity through a non-linear optoelectronic feedback loop. We establish experimentally the excitable character of a single node, which stems from the slow-fast nature of the system, adequately described by a set of rate equations with three well separated time scales. Beyond the excitable regime, the system undergoes relaxation oscillations where the nodes display canard dynamics. We show numerically that, without noise, the coupled system follows an intricate canard trajectory, with the nodes switching on one by one. While incorporating noise leads to a better correspondence between numerical simulations and experimental data, it also has an unexpected ordering effect on the canard orbit, causing the nodes to switch on closer together in time. We find that the dispersion of the trajectories of the network nodes in phase space is minimized for a non-zero noise strength, and call this phenomenon canard resonance.

Comparison of Different Spike Train Synchrony Measures Regarding Their Robustness to Erroneous Data From Bicuculline-Induced Epileptiform Activity

As synchronized activity is associated with basic brain functions and pathological states, spike train synchrony has become an important measure to analyze experimental neuronal data. Many measures of spike train synchrony have been proposed, but there is no gold standard allowing for comparison of results from different experiments. This work aims to provide guidance on which synchrony measure is best suited to quantify the effect of epileptiform-inducing substances (e.g., bicuculline, BIC) in in vitro neuronal spike train data. Spike train data from recordings are likely to suffer from erroneous spike detection, such as missed spikes (false negative) or noise (false positive). Therefore, different timescale-dependent (cross-correlation, mutual information, spike time tiling coefficient) and timescale-independent (Spike-contrast, phase synchronization (PS), A-SPIKE-synchronization, A-ISI-distance, ARI-SPIKE-distance) synchrony measures were compared in terms of their robustness to erroneous spike trains. For this purpose, erroneous spike trains were generated by randomly adding (false positive) or deleting (false negative) spikes (in silico manipulated data) from experimental data. In addition, experimental data were analyzed using different spike detection threshold factors in order to confirm the robustness of the synchrony measures. All experimental data were recorded from cortical neuronal networks on microelectrode array chips, which show epileptiform activity induced by the substance BIC. As a result of the in silico manipulated data, Spike-contrast was the only measure that was robust to false-negative as well as false-positive spikes. Analyzing the experimental data set revealed that all measures were able to capture the effect of BIC in a statistically significant way, with Spike-contrast showing the highest statistical significance even at low spike detection thresholds. In summary, we suggest using Spike-contrast to complement established synchrony measures because it is timescale independent and robust to erroneous spike trains.

Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons

We consider the effect of small independent local noise on a network of quadratic integrate-and-fire neurons, globally coupled via synaptic pulses of finite width. The Fokker-Planck equation for a network of infinite size is reduced to a low-dimensional system of ordinary differential equations using the recently proposed perturbation theory based on circular cumulants. A bifurcation analysis of the reduced equations is performed, and areas in the parameter space, where the noise causes macroscopic oscillations of the network, are determined. The validity of the reduced equations is verified by comparing their solutions with "exact" solutions of the Fokker-Planck equation, as well as with the results of direct simulation of stochastic microscopic dynamics of a finite-size network.

Control of coherence resonance by self-induced stochastic resonance in a multiplex neural network

We consider a two-layer multiplex network of diffusively coupled FitzHugh-Nagumo (FHN) neurons in the excitable regime. We show that the phenomenon of coherence resonance (CR) in one layer can not only be controlled by the network topology, the intra- and interlayer time-delayed couplings, but also by another phenomenon, namely, self-induced stochastic resonance (SISR) in the other layer. Numerical computations show that when the layers are isolated, each of these noise-induced phenomena is weakened (strengthened) by a sparser (denser) ring network topology, stronger (weaker) intralayer coupling forces, and longer (shorter) intralayer time delays. However, CR shows a much higher sensitivity than SISR to changes in these control parameters. It is also shown, in contrast to SISR in a single isolated FHN neuron, that the maximum noise amplitude at which SISR occurs in the network of coupled FHN neurons is controllable, especially in the regime of strong coupling forces and long time delays. In order to use SISR in the first layer of the multiplex network to control CR in the second layer, we first choose the control parameters of the second layer in isolation such that in one case CR is poor and in another case, nonexistent. It is then shown that a pronounced SISR can not only significantly improve a poor CR, but can also induce a pronounced CR, which was nonexistent in the isolated second layer. In contrast to strong intralayer coupling forces, strong interlayer coupling forces are found to enhance CR, while long interlayer time delays, just as long intralayer time delays, deteriorate CR. Most importantly, we find that in a strong interlayer coupling regime, SISR in the first layer performs better than CR in enhancing CR in the second layer. But in a weak interlayer coupling regime, CR in the first layer performs better than SISR in enhancing CR in the second layer. Our results could find novel applications in noisy neural network dynamics and engineering.

How synaptic plasticity influences spike synchronization and its transitions in complex neuronal network

There is evidence that synaptic plasticity is a vital feature of realistic neuronal systems. This study, describing synaptic plasticity by a modified Oja learning rule, focuses on the effect of synapse learning rate on spike synchronization and its relative transitions in a Newman-Watts small-world neuronal network. The individual dynamics of each neuron is modeled by a simple Rulkov map that produces spiking behavior. Numerical results have indicated that large coupling can lead to a spatiotemporally synchronous pattern of spiking neurons; in addition, this kind of spike synchronization can emerge intermittently by turning information transmission delay between coupled neurons. Interestingly, with the advent of synaptic plasticity, spike synchronization is gradually destroyed by increasing synapse learning rate; moreover, the phenomenon of intermittent synchronization transitions becomes less and less obvious and it even disappears for relative larger learning rate. Further simulations confirm that spike synchronization as well as synchronization transitions is largely independent of network size. Meanwhile, we detect that large shortcuts probability can facilitate spike synchronization, but it is disadvantageous for delay-induced synchronization transitions.

Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

Weak multiplexing induces coherence resonance

Using the model of a FitzHugh-Nagumo system in the excitable regime, we study the impact of multiplexing on coherence resonance in a two-layer network. We show that multiplexing allows for the control of the noise-induced dynamics. In particular, we find that multiplexing induces coherence resonance in networks that do not demonstrate this phenomenon in isolation. Examples are provided by deterministic networks and networks where the strength of interaction between the elements is not optimal for coherence resonance. In both cases, we show that the control strategy based on multiplexing can be successfully applied even for weak coupling between the layers. Moreover, for the case of deterministic networks, we obtain a counter-intuitive result: the multiplex-induced coherence resonance in the layer which is deterministic in isolation manifests itself even more strongly than that in the noisy layer.

Effect of spike-timing-dependent plasticity on stochastic burst synchronization in a scale-free neuronal network

We consider an excitatory population of subthreshold Izhikevich neurons which cannot fire spontaneously without noise. As the coupling strength passes a threshold, individual neurons exhibit noise-induced burstings. This neuronal population has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). However, STDP was not considered in previous works on stochastic burst synchronization (SBS) between noise-induced burstings of sub-threshold neurons. Here, we study the effect of additive STDP on SBS by varying the noise intensity D in the Barabási-Albert scale-free network (SFN). One of our main findings is a Matthew effect in synaptic plasticity which occurs due to a positive feedback process. Good burst synchronization (with higher bursting measure) gets better via long-term potentiation (LTP) of synaptic strengths, while bad burst synchronization (with lower bursting measure) gets worse via long-term depression (LTD). Consequently, a step-like rapid transition to SBS occurs by changing D, in contrast to a relatively smooth transition in the absence of STDP. We also investigate the effects of network architecture on SBS by varying the symmetric attachment degree [Formula: see text] and the asymmetry parameter [Formula: see text] in the SFN, and Matthew effects are also found to occur by varying [Formula: see text] and [Formula: see text]. Furthermore, emergences of LTP and LTD of synaptic strengths are investigated in details via our own microscopic methods based on both the distributions of time delays between the burst onset times of the pre- and the post-synaptic neurons and the pair-correlations between the pre- and the post-synaptic instantaneous individual burst rates (IIBRs). Finally, a multiplicative STDP case (depending on states) with soft bounds is also investigated in comparison with the additive STDP case (independent of states) with hard bounds. Due to the soft bounds, a Matthew effect with some quantitative differences is also found to occur for the case of multiplicative STDP.

Stochastic spike synchronization in a small-world neural network with spike-timing-dependent plasticity

We consider the Watts-Strogatz small-world network (SWN) consisting of subthreshold neurons which exhibit noise-induced spikings. This neuronal network has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). In previous works without STDP, stochastic spike synchronization (SSS) between noise-induced spikings of subthreshold neurons was found to occur in a range of intermediate noise intensities. Here, we investigate the effect of additive STDP on the SSS by varying the noise intensity. Occurrence of a "Matthew" effect in synaptic plasticity is found due to a positive feedback process. As a result, good synchronization gets better via long-term potentiation of synaptic strengths, while bad synchronization gets worse via long-term depression. Emergences of long-term potentiation and long-term depression of synaptic strengths are intensively investigated via microscopic studies based on the pair-correlations between the pre- and the post-synaptic IISRs (instantaneous individual spike rates) as well as the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, the effects of multiplicative STDP (which depends on states) on the SSS are studied and discussed in comparison with the case of additive STDP (independent of states). These effects of STDP on the SSS in the SWN are also compared with those in the regular lattice and the random graph.

Coherence resonance in a network of FitzHugh-Nagumo systems: Interplay of noise, time-delay, and topology

We systematically investigate the phenomena of coherence resonance in time-delay coupled networks of FitzHugh-Nagumo elements in the excitable regime. Using numerical simulations, we examine the interplay of noise, time-delayed coupling, and network topology in the generation of coherence resonance. In the deterministic case, we show that the delay-induced dynamics is independent of the number of nearest neighbors and the system size. In the presence of noise, we demonstrate the possibility of controlling coherence resonance by varying the time-delay and the number of nearest neighbors. For a locally coupled ring, we show that the time-delay weakens coherence resonance. For nonlocal coupling with appropriate time-delays, both enhancement and weakening of coherence resonance are possible.

Control of in vivo ictogenesis via endogenous synaptic pathways

The random nature of seizures poses difficult challenges for epilepsy research. There is great need for a reliable method to control the pathway to seizure onset, which would allow investigation of the mechanisms of ictogenesis and optimization of treatments. Our hypothesis is that increased random afferent synaptic activity (i.e. synaptic noise) within the epileptic focus is one endogenous method of ictogenesis. Building upon previous theoretical and in vitro work showing that synaptic noise can induce seizures, we developed a novel in vivo model of ictogenesis. By increasing the excitability of afferent connections to the hippocampus, we control the risk of temporal lobe seizures during a specific time period. The afferent synaptic activity in the hippocampus was modulated by focal microinjections of potassium chloride into the nucleus reuniens, during which the risk of seizure occurrence increased substantially. The induced seizures were qualitatively and quantitatively indistinguishable from spontaneous ones. This model thus allows direct control of the temporal lobe seizure threshold via endogenous pathways, providing a novel tool in which to investigate the mechanisms and biomarkers of ictogenesis, test for seizure threshold, and rapidly tune antiseizure treatments.

Effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks

In this paper, we study effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks. Our focus is on the impact of two parameters, namely the time delay τ and the probability of partial time delay p_delay, whereby the latter determines the probability with which a connection between two neurons is delayed. Our research reveals that partial time delays significantly affect phase synchronization in this system. In particular, partial time delays can either enhance or decrease phase synchronization and induce synchronization transitions with changes in the mean firing rate of neurons, as well as induce switching between synchronized neurons with period-1 firing to synchronized neurons with period-2 firing. Moreover, in comparison to a neuronal network where all connections are delayed, we show that small partial time delay probabilities have especially different influences on phase synchronization of neuronal networks.

Impacts of clustering on noise-induced spiking regularity in the excitatory neuronal networks of subnetworks

In this paper, we investigate how clustering factors influent spiking regularity of the neuronal network of subnetworks. In order to do so, we fix the averaged coupling probability and the averaged coupling strength, and take the cluster number M, the ratio of intra-connection probability and inter-connection probability R, the ratio of intra-coupling strength and inter-coupling strength S as controlled parameters. With the obtained simulation results, we find that spiking regularity of the neuronal networks has little variations with changing of R and S when M is fixed. However, cluster number M could reduce the spiking regularity to low level when the uniform neuronal network's spiking regularity is at high level. Combined the obtained results, we can see that clustering factors have little influences on the spiking regularity when the entire energy is fixed, which could be controlled by the averaged coupling strength and the averaged connection probability.

Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons

We are interested in noise-induced firings of subthreshold neurons which may be used for encoding environmental stimuli. Noise-induced population synchronization was previously studied only for the case of global coupling, unlike the case of subthreshold spiking neurons. Hence, we investigate the effect of complex network architecture on noise-induced synchronization in an inhibitory population of subthreshold bursting Hindmarsh-Rose neurons. For modeling complex synaptic connectivity, we consider the Watts-Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization. Thus, noise-induced burst synchronization (synchrony on the slow bursting time scale) and spike synchronization (synchrony on the fast spike time scale) are found to appear in a synchronized region of the [Formula: see text]-[Formula: see text] plane ([Formula: see text]: synaptic inhibition strength and [Formula: see text]: noise intensity). As the rewiring probability [Formula: see text] is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the [Formula: see text]-[Formula: see text] plane, while the region of the burst synchronization decreases slowly. We separate the slow bursting and the fast spiking time scales via frequency filtering, and characterize the noise-induced burst and spike synchronizations by employing realistic order parameters and statistical-mechanical measures introduced in our recent work. Thus, the bursting and spiking thresholds for the burst and spike synchronization transitions are determined in terms of the bursting and spiking order parameters, respectively. Furthermore, we also measure the degrees of burst and spike synchronizations in terms of the statistical-mechanical bursting and spiking measures, respectively.

Clustering versus non-clustering phase synchronizations

Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled systems.

Periodic coupling strength-dependent multiple coherence resonance by time delay in Newman-Watts neuronal networks

Recently, multiple coherence resonance induced by time delay has been observed in neuronal networks with constant coupling strength. In this paper, by employing Newman-Watts Hodgkin-Huxley neuron networks with time-periodic coupling strength, we study how the temporal coherence of spiking behavior and coherence resonance by time delay change when the frequency of periodic coupling strength is varied. It is found that delay induced coherence resonance is dependent on periodic coupling strength and increases when the frequency of periodic coupling strength increases. Periodic coupling strength can also induce multiple coherence resonance, and the coherence resonance occurs when the frequency of periodic coupling strength is approximately multiple of the spiking frequency. These results show that for periodic coupling strength time delay can more frequently optimize the temporal coherence of spiking activity, and periodic coupling strength can repetitively optimize the temporal coherence of spiking activity as well. Frequency locking may be the mechanism for multiple coherence resonance induced by periodic coupling strength. These findings imply that periodic coupling strength is more efficient for enhancing the temporal coherence of spiking activity of neuronal networks, and thus it could play a more important role in improving the time precision of information processing and transmission in neural networks.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

Impact of connection delays on noise-induced spatiotemporal patterns in neuronal networks

In the present work, we investigate the nontrivial roles of independent Gaussian noise and time-delayed coupling on the synchronous dynamics and coherence property of Fitz Hugh-Nagumo neurons on small-world networks by numerical simulations. First, it is shown that an intermediate level of noise in the neuronal networks can optimally induce a temporal coherence state when the delay in the coupling is absent. We find that this phenomenon is robust to changes of the coupling strength and the rewiring probability of small-world networks. Then, when appropriately tuned delays with moderate values are included in the coupling, the neurons on the networks can reach higher ordered spatiotemporal patterns which are the most coherent in time and almost synchronized in space. Moreover, the tuned delays are within a range, and the period of the firing activity is delay-dependent which equals nearly to the length of the coupling delay. This result implies that the higher ordered spatiotemporal dynamics induced by intermediate delays could be the result of a locking between the period-1 neuronal spiking activity and the delay. The performance of moderate delays in enhancing the ordered spatiotemporal patterns is also examined to be robust against variations of the network randomness.

Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol oscillators near a codimension-two bifurcation point

Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincaré map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves.

Propagation of spiking regularity and double coherence resonance in feedforward networks

We investigate the propagation of spiking regularity in noisy feedforward networks (FFNs) based on FitzHugh-Nagumo neuron model systematically. It is found that noise could modulate the transmission of firing rate and spiking regularity. Noise-induced synchronization and synfire-enhanced coherence resonance are also observed when signals propagate in noisy multilayer networks. It is interesting that double coherence resonance (DCR) with the combination of synaptic input correlation and noise intensity is finally attained after the processing layer by layer in FFNs. Furthermore, inhibitory connections also play essential roles in shaping DCR phenomena. Several properties of the neuronal network such as noise intensity, correlation of synaptic inputs, and inhibitory connections can serve as control parameters in modulating both rate coding and the order of temporal coding.

Multiple coherence resonance induced by time-periodic coupling in stochastic Hodgkin-Huxley neuronal networks

In this paper, we study the effect of time-periodic coupling strength (TPCS) on the spiking coherence of Newman-Watts small-world networks of stochastic Hodgkin-Huxley (HH) neurons and investigate the relations between the coupling strength and channel noise when coherence resonance (CR) occurs. It is found that, when the amplitude of TPCS is varied, the spiking induced by channel noise can exhibit CR and coherence bi-resonance (CBR), and the CR moves to a smaller patch area (bigger channel noise) when the amplitude increases; when the frequency of TPCS is varied, the intrinsic spiking can exhibit CBR and multiple CR, and the CR always occurs when the frequency is equal to or multiple of the spiking period, manifesting as the locking between the frequencies of the intrinsic spiking and the coupling strength. These results show that TPCS can greatly enhance and optimize the intrinsic spiking coherence, and favors the spiking with bigger channel noise to exhibit CR. This implies that, compared to constant coupling strength, TPCS may play a more efficient role for improving the time precision of the information processing in stochastic neuronal networks.

Using Jensen's inequality to explain the role of regular calcium oscillations in protein activation

Oscillations of cytosolic Ca(2 +) are very important for cellular signalling in excitable and non-excitable cells. The information of various extracellular stimuli is encoded into oscillating patterns of Ca(2 +) that subsequently lead to the activation of different Ca(2 +)-sensitive target proteins in the cell. The question remains, however, why this information is transmitted by means of an oscillating rather than a constant signal. Here we show that, in fact, Ca(2 +) oscillations can achieve a better activation of target proteins than a comparable constant signal with the same amount of Ca(2 +) used. For this we use Jensen's inequality that describes the relation between the function value of the average of a set of argument values and the average of the function values of the arguments from that set. We analyse the role of the cooperativity of the binding of Ca(2 +) and of zero-order ultrasensitivity, which are two properties that are often observed in experiments on the activation of Ca(2 +)-sensitive target proteins. Our results apply to arbitrary oscillation shapes and a very general decoding model, thus generalizing the observations of several previous studies. We compare our results with data from experimental studies investigating the activation of nuclear factor of activated T cells (NFAT) and Ras by oscillatory and constant signals. Although we are restricted to specific approximations due to the lack of detailed kinetic data, we find good qualitative agreement with our theoretical predictions.

Control of noise-induced coherent behaviors in an array of excitable elements by time-delayed feedback

We investigate feedback-controlled coherent dynamics in a two-dimensional array of excitable elements. We demonstrate that one can freely enhance or reduce the spatiotemporal coherence of noise-induced oscillation, such as coherence resonance and phase synchronization, by controlling both the delay time and the feedback gain. Furthermore, we find that noise-induced oscillations are entrained by the feedback force in a certain range of the delay time. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator model.

Synaptic noise and physiological coupling generate high-frequency oscillations in a hippocampal computational model

There is great interest in the role of coherent oscillations in the brain. In some cases, high-frequency oscillations (HFOs) are integral to normal brain function, whereas at other times they are implicated as markers of epileptic tissue. Mechanisms underlying HFO generation, especially in abnormal tissue, are not well understood. Using a physiological computer model of hippocampus, we investigate random synaptic activity (noise) as a potential initiator of HFOs. We explore parameters necessary to produce these oscillations and quantify the response using the tools of stochastic resonance (SR) and coherence resonance (CR). As predicted by SR, when noise was added to the network the model was able to detect a subthreshold periodic signal. Addition of basket cell interneurons produced two novel SR effects: 1) improved signal detection at low noise levels and 2) formation of coherent oscillations at high noise that were entrained to harmonics of the signal frequency. The periodic signal was then removed to study oscillations generated only by noise. The combined effects of network coupling and synaptic noise produced coherent, periodic oscillations within the network, an example of CR. Our results show that, under normal coupling conditions, synaptic noise was able to produce gamma (30-100 Hz) frequency oscillations. Synaptic noise generated HFOs in the ripple range (100-200 Hz) when the network had parameters similar to pathological findings in epilepsy: increased gap junctions or recurrent synaptic connections, loss of inhibitory interneurons such as basket cells, and increased synaptic noise. The model parameters that generated these effects are comparable with published experimental data. We propose that increased synaptic noise and physiological coupling mechanisms are sufficient to generate gamma oscillations and that pathologic changes in noise and coupling similar to those in epilepsy can produce abnormal ripples.

From puffs to global Ca2+ signals: how molecular properties shape global signals

The universality of Ca(2+) as second messenger in living cells is achieved by a rich spectrum of spatiotemporal cellular concentration dynamics. Ca(2+) release from internal storage compartments plays a key role in shaping cytosolic Ca(2+) signals. Deciphering this signaling mechanism is essential for a deeper understanding of its physiological function and general concepts of cell signaling. Here, we review recent experimental findings demonstrating the stochasticity of Ca(2+) oscillations and its relevance for modeling Ca(2+) dynamics. The stochasticity arises by the hierarchical signal structure that carries molecular fluctuations of single channels onto the level of the cell leading to a stochastic medium as theoretically predicted. The result contradicts the current opinion of Ca(2+) being a cellular oscillator. We demonstrate that cells use array enhanced coherence resonance to form rather regular spiking signals and that the "oscillations" carry information despite the involved stochasticity. The knowledge on the underlying mechanism also allows for determination of intrinsic properties from global observations. In the second part of the paper, we briefly survey different modeling approaches with regard to the experimental results. We focus on the dependence of the standard deviation on the mean period of the oscillations. It shows that limit cycle oscillations cannot describe the experimental data and that generic models have to include the spatial aspects of Ca(2+) signaling.

Stochastic hierarchical systems: excitable dynamics

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.

The selectivity of noise and coupling for coherence biresonance and array-enhanced coherence biresonance in coupled neural systems

The selectivity of noise and coupling for coherence biresonance (CBR) and array-enhanced coherence biresonance (AECBR) in coupled neural systems has been investigated. It is shown that, depending on the coupling strength and noise intensity, various coherence behaviors and phenomena are exhibited, including CBR, coherence resonance without tuning, AECBR and undamped signal transmission. There exist optimal coupling and noise regions for the occurrence of CBR and AECBR in the transmission of noise-induced oscillations (NIOs).

Investigating the properties of self-organization and synchronization in electronic systems

Nonlinear cooperative behavior appears naturally in many systems, such as cardiac cell oscillations; cellular calcium oscillations; oscillatory chemical reactions, and fireflies. Such systems have been studied in detail due to their inherent properties of robustness, adaptability, scalability, and emergence. In this paper, such nonlinear cooperative behaviors are considered within the domain of electronic system design. We investigate these desirable properties in a system composed of electronic oscillators. The paper presents a series of circuit simulation results showing that self-organizing principles, which can be emulated in an electronic circuit, enable the systems to show a phase transition to synchronization, in a manner similar to those of natural systems. Circuit simulation results presented here show that the circuits are robust to the unreliable performance of the electronic oscillators and tolerant to their run-time faults. These are important findings for future engineering applications in which the system's elements are likely to be unreliable and faulty, such as in molecular- and nanoelectronic systems.

Coherence resonance induced by rewiring in complex networks

We report a novel coherent excitation phenomenon in a heterogeneous network of coupled FitzHugh-Nagumo elements. It is demonstrated that dynamical rewiring in the network can play a constructive role to bring on coherent excitations. The coherence factor as the function of rewiring time interval represents a nontrivial phenomenon which is a fingerprint of coherence resonance. We call this resonant behavior caused by dynamical wiring changes the network-rewiring-induced coherence resonance. The mechanism can be understood by the effective noise played by the rewiring process.

Optimal spatial synchronization on scale-free networks via noisy chemical synapses

We show that the spatial synchronization of noise-induced excitations on scale-free networks, mediated through nonlinear chemical coupling, depends vitally on the intensity of additive noise and the coupling strength. In particular, a twofold optimization is needed for achieving maximal spatial synchrony, thus indicating the existence of an optimal noise intensity as well as an optimal coupling strength. On the other hand, the traditional linear coupling via gap junctions, while still requiring a fine-tuning of the noise intensity, does not postulate the existence of an optimal coupling strength since the synchronization increases monotonously with the increasing coupling strength. Presented results reveal inherent differences in optimal spatial synchronization evoked by chemical and electrical coupling, and could hence help to pinpoint their specific roles in networked systems.

Coherence, collective rhythm, and phase difference distribution in populations of stochastic genetic oscillators with cellular communication

An ensemble of stochastic genetic relaxation oscillators via phase-attractive or repulsive cell-to-cell communication are investigated. In the phase-attractive coupling case, it is found that cellular communication can enhance self-induced stochastic resonance as well as collective rhythms, and that different intensities of noise resulting from the fluctuation of intrinsic chemical reactions or the extrinsic environment can induce stochastic limit cycles with different amplitudes for a large cell density. In contrast, in the phase-repulsive coupling case, the distribution of phase differences among the stochastic oscillators can display such characteristic as unimodality, bimodality or polymodality, depending on both noise intensity and cell number, but the modality of phase difference distribution almost keeps invariant for an arbitrary noise intensity as the cell number is beyond a threshold.

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.

Control of spiking regularity in a noisy complex neural network

The effects of spatiotemporally correlated noise on the regularity of spiking oscillations are studied in a network composed of Fitz-Hugh-Nagumo neurons. The spiking regularity of the neural network becomes the best at a moderate noise intensity, indicating the occurrence of coherence resonance (CR). The CR in a Watts-Strogatz small-world network is further improved by adding a small fraction of long-range connections. Given a set of temporal correlation constant tau and spatial correlation length lambda of the noise, there exists an optimal network topology randomness, at which the spiking oscillations show the best regularity. The optimal randomness of the network topology at different tau and lambda varies in a narrow range. Changing lambda does not affect the optimal tau for achieving the most regular spike train, whereas varying tau, the best spiking regularity emerges at different optimal lambda.

Diversity-induced coherence resonance in spatially extended chaotic systems

The effect of parameter diversity on coupled Chua systems is investigated. In the absence of diversity, the systems jump back and forth between two variable domains of a chaotic attractor, and the residence times within a single domain are uncertain. By introducing parameter diversity, a combined numerical and analytical approach indicates that the systems can jump regularly from one domain to another at an intermediate range of diversity, a signature of coherence resonance. Furthermore, the influences of coupling strength and the number of units are also considered. Our results provide a possibility for the control of chaos in spatially extended chaotic systems by the manipulation of parameter diversity.

Interacting stochastic oscillators

Stochastic coherence (SC) and self-induced stochastic resonance (SISR) are two distinct mechanisms of noise-induced coherent motion. For interacting SC and SISR oscillators, we find that whether or not phase synchronization is achieved depends sensitively on the coupling strength and noise intensities. Specifically, in the case of weak coupling, individual oscillators are insensitive to each other, whereas in the case of strong coupling, one fixed oscillator with optimal coherence can be entrained to the other, adjustable oscillator (i.e., its noise intensity is tunable), achieving phase-locking synchronization, as long as the tunable noise intensity is not beyond a threshold; such synchronization is lost otherwise. For an array lattice of SISR oscillators, except for coupling-enhanced coherence similar to that found in the case of coupled SC oscillators, there is an optimal network topology degree (i.e., number of coupled nodes), such that coherence and synchronization are optimally achieved, implying that the system-size resonance found in an ensemble of noise-driven bistable systems can occur in coupled SISR oscillators.

Array-enhanced coherence resonance and phase synchronization in a two-dimensional array of excitable chemical oscillators

We investigate the spatiotemporal dynamics in a two-dimensional array of excitable elements subjected to independent external noise, where elements are prepared by localizing the Belousov-Zhabotinsky reaction in a gel matrix. We experimentally demonstrate that the coherence of noise-induced firings is improved with increasing the array size, i.e., the occurrence of array-enhanced coherence resonance. Furthermore, it is found that synchronization among oscillators which are barely coupled can be achieved via coherence resonance. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator reaction-diffusion model.

Non-Markovian approach to globally coupled excitable systems

We consider stochastic excitable units with three discrete states. Each state is characterized by a waiting time density function. This approach allows for a non-Markovian description of the dynamics of separate excitable units and of ensembles of such units. We discuss the emergence of oscillations in a globally coupled ensemble with excitatory coupling. In the limit of a large ensemble we derive the non-Markovian mean-field equations: nonlinear integral equations for the populations of the three states. We analyze the stability of their steady solutions. Collective oscillations are shown to persist in a large parameter region beyond supercritical and subcritical Hopf bifurcations. We compare the results with simulations of discrete units as well as of coupled FitzHugh-Nagumo systems.

Structural-diversity-enhanced cellular ability to detect subthreshold extracellular signals

We study the influence of the structural diversity of cells on subthreshold agonist signals in coupled hepatocytes systems. The variance of the cellular structural parameter sigma is employed to characterize the structural diversity. It is found that structural diversity enhances the cellular ability to detect extracellular weak signals through intracellular Ca2+ oscillations and the regularity of Ca2+ spikes undergoes a maximum with a variation of sigma, indicating the occurrence of structural-diversity-induced coherence resonance. Furthermore, the effects of the level of subthreshold stimulus and junctional coupling strength on the behavior of Ca2+ dynamics are also considered. Analysis indicates that these phenomena have inherent relevance to both the bifurcation feature of a single cell and intercellular interaction through junctional coupling. Our findings may exhibit that structural diversity plays a constructive role in biological systems.

Mean-field coupling of calcium oscillations in a multicellular system of rat hepatocytes

In a multicellular system of rat hepatocytes and even in an intact liver, cytoplasmic calcium oscillations are synchronized and highly coordinated. In this paper, the mean-field coupling term has been introduced to describe the coupling flux, which is more efficient than gap junctional coupling terms. An optimal coupling strength and an optimal stimulation level for the synchronization of the coupled system have been observed in this paper. Moreover, it has been proved that these results are independent of the cells number. Interestingly, it has been observed that the intracellular noise and the extracellular noise have different effects on the synchronization of the coupled system.

Coupling effects on energy transduction in coupled polymer chains with perturbation of noise

Noise-assistant transduction was investigated in coupled polymer chains where one subsystem was exposed to environment noise. It was found that coupling could transfer oscillation from one subsystem disturbed by noise to the other not disturbed by noise and play a role of a noise filtering for the other. Then, a sort of coupling-induced synchronization was investigated as a function of noise intensity and coupling strength. In particular, we calculated the minimum coupling strength to reach synchronization and pointed out that noise dominated at small coupling strength, otherwise, coupling dominated.

Noise accelerates synchronization of coupled nonlinear oscillators

For a chain of homogeneous nonlinear oscillators starting from different initial phases, a certain amount of time is required for the system to evolve to complete phase synchronization. The effect of independent noise in such a system was investigated, and an optimal noise intensity was found that minimized the average synchronization time. Both threshold noise and connection noise show similar effects. The features of the phenomenon and the underlying mechanism are discussed through the analysis of a two-unit system and the numerical studies of chains up to 30 units in length.